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Revision:Vectors 1

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Vectors

A vector quantity has both length (magnitude) and direction. The opposite is a scalar quantity, which only has magnitude. Vectors can be denoted by AB, a, or AB (with an arrow above the letters).

If:

\displaystyle \mathbf{a} = \begin{pmatrix}
  3  \\
  2  
\end{pmatrix}
then the vector will look as follows:



(Diagram missing)



NB1: When writing vectors as one number above another in brackets, this is known as a column vector.

NB2: in textbooks and here, vectors are indicated by bold type. However, when you write them, you need to put a line underneath the vector to indicate it (i.e. a).


Multiplication by a Scalar

When multiplying a vector by a scalar (i.e. a number), multiply each component of the vector by the scalar.


Example

If \displaystyle \mathbf{a} = \begin{pmatrix}
  3 \\
  2 
\end{pmatrix}, and \displaystyle \mathbf{b} = 2\mathbf{a}, sketch a and b.


If \displaystyle \mathbf{a} = \begin{pmatrix}
  3 \\
  2 
\end{pmatrix}, \displaystyle \mathbf{b} = 2\mathbf{a} = \begin{pmatrix}
  6 \\
  4 
\end{pmatrix}.



(Diagram missing)



Vector Manipulation

(Diagram missing)



Example

If If \displaystyle \mathbf{a} = \begin{pmatrix}
  -5 \\
  3 
\end{pmatrix} and If \displaystyle \mathbf{b} = \begin{pmatrix}
  2 \\
  1 
\end{pmatrix}, find the magnitude of their resultant.


The resultant of two or more vectors is their sum.

The resultant therefore is \displaystyle \begin{pmatrix}
  -3 \\
  4 
\end{pmatrix}.


The magnitude of this is \displaystyle \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5.


The addition and subtraction of vectors can be shown diagrammatically. To find a + b, draw a and then draw b at the end of a. The resultant is the line between the start of a and the end of b.

To find a - b, find -b (see above) and add this to a.


Example

(Diagram missing)



Unit Vectors

A unit vector has a magnitude of 1. The unit vector in the direction of the x-axis is i and the unit vector in the direction of the y-axis is j. For example on a graph, 3i + 4j would be at (3 , 4). This method is another method of writing down vectors.

Example

(3i + j) + (5i - 4j) = 8i - 3j.


This is equivalent to:

\displaystyle \begin{pmatrix}
  3 \\
  5 
\end{pmatrix} +\begin{pmatrix}
  5 \\
  -4 
\end{pmatrix} = \begin{pmatrix}
  8 \\
  -3 
\end{pmatrix}.


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